Molecular Modeling in Engineering: Methods and Applications

1. Nikolai V. Priezjev
Department of Mechanical Engineering
Michigan State University
Molecular Modeling in
Engineering: Methods and Applications
Lecture: Ensembles and free energy in
Monte Carlo simulations

2. Department of Mechanical Engineering Michigan State University
Overview of the previous lecture
Monte Carlo Method: (NVT Canonical Ensemble)
• Goal: to generate molecular configurations (states) with
Boltzmann distribution P(E)~exp(-E/kT)
• Ensemble average: Pressure, Energy, CV
  

M
i
PPPP
M
P 1 4321 ...
1
2
22
2
2
)(
P
PP
P
P 


kP
iteration k=1, 2…
P
Ensembles for Monte Carlo simulations: Chapter 5
1. Canonical Ensemble (NVT)
2. Isobaric-Isothermal (NPT)
3. Microcanonical (NVE)
4. Grand Canonical Ensemble (μVT)
μ-chemical potential
The ensembles are equivalent; question of convenience.

3. Statistical Mechanics 101
Boltzmann distribution: the probability to find a system in a state i with energy Ei
kTE
i
i
eEP /
~)( 
Partition function:
kTE
i
i
i
kTE ii
egeZ // 
 
degeneracy =
# of states with energy
ig
iE
kTE
i
i
e
Z
EP /1
)( 

kT
1

Helmholtz free energy: ZkTTVNF ln),,( 
Then, the internal energy:
 d
Zd
d
dZ
Z
eE
Z
E
i
kTE
i
i
ln11 /
  
canonical
Specific heat:
Gibbs free energy: ZkTTPNF ln),,( 
isothermal-isobaric

4. Name All states of: Probability distribution Schematic
Microcanonical
(EVN)
given EVN 1
i 

Canonical
(TVN)
all energies 1( ) iE
i Q
E e 
 

Isothermal-isobaric
(TPN)
all energies and
volumes
( )1( , ) i iE PV
i iE V e 
  


Grand-canonical
(TV)
all energies and
molecule numbers
( )1( , ) i iE N
i iE N e  
  


Ensemble Thermodynamic Potential Partition Function Bridge Equation
Microcanonical Entropy, S
1   / ln ( , , )S k E V N 
Canonical Helmholtz, A
iE
Q e

 
ln ( , , )A Q T V N 
Isothermal-isobaric Gibbs, G ( )i iE PV
e
 
  
ln ( , , )G T P N  
Grand-canonical Hill, L = –PV ( )i iE N
e
  
  
ln ( , , )PV T V  

5. Density of states versus Boltzmann factor
Degeneracy E(A) = E(B)
then P(A)=P(B)~exp(-E/kT)
If E(A) > E(B) then
density of states N(A)>N(B)
A B
A B
Potential energy E (sum VLJ)
Probabilityofstate
N(E) =
density
of states
P(E)~exp(-E/kT)
Free energy then:
)(ln EHistkTF 
kTE
i
i
i
kTE ii
egeZ // 
 
Partition function:

6. Nematic Isotropic
(ordered) (disordered)
The usual Metropolis
Monte Carlo move:
Department of Mechanical Engineering Michigan State University
Cluster Monte
Carlo move
Analogy with
LJ potential.

7. (Uniaxal) 3D orientational order parameter in liquid crystals
Department of Mechanical Engineering Michigan State University
Isotropic phaseNematic phase
No positional but
orientational order!
director
Discontinuous jump of the order parameter
at the critical temperature = first order phase
transition (coexistence of two phases)
Isotropic (disordered) phase

8. Free energy at the nematic-isotropic transition:
Canonical Ensemble: (N, V, T)
NI
Freeenergy
Potential energy –E = sum over all nn pairs/N
i.e. N →inf; or L-3 → 0
L=70
L=70
TransitiontemperatureTc(L)
60
L=60
Notations: L=system size; V=L3 =volume; N=number of molecules
Department of Mechanical Engineering Michigan State University
L3
50
30
L=50
L=40
Histogram reweighting [from T(L) to Tc(L)]
L=30

9. In Wang-Landau, g(E) is initially set to 1 and modified “on the fly”. Monte
Carlo moves are accepted with probability
Each time when an energy level E is visited, its density of states is
updated by a modification factor f >1, i.e.,
Observation: if a random walk is performed with probability
proportional to reciprocal of density of states
then a flat energy histogram could be obtained.
Wang-Landau Monte Carlo Method






 1,
)(
)(
min)(
2
1
21
Eg
Eg
EEp
)(/1)( EgEp 
The density of states is not known a priori.
fEgEg )()( 
Fugao Wang and D. P. Landau "Efficient, Multiple-Range Random Walk Algorithm to Calculate
the Density of States", Physical Review Letters 86 pp. 2050-2053 (2001). Times Cited: 741

10. Advantages:
1. Simple formulation and general applicability;
2. Entropy and free energy information derivable from g(E);
3. Each energy state is visited with equal probability, so energy barriers are
overcome with relative ease.
kTE
i
i
i
kTE ii
egeZ // 
 
Example: Two-dimensional Ising Model
- +
+
+
+
+
+
+
+
+
+
+
+
++
+
+
-
-
--
-
- -
- -
-
- --
-
-
---
- The energy of
configuration σ is
E(σ) = - J ∑<ij> σi σj
where i and j run over a
lattice, <ij> denotes
nearest neighbors, σ = ±1
σ = {σ1, σ2, …, σi, … }
Wang-Landau Monte Carlo Method

11. D. P. Landau, Shan-Ho Tsai, and M. Exler "A new approach to Monte Carlo simulations in statistical
physics: Wang-Landau sampling", American Journal of Physics 72 pp. 1294-1302 (2004).
Free energy:
  2
22
22
/)/(
Tk
UU
TkUTUC
B
B
NVT
NVTV

 
Specific heat:
Wang-Landau Monte Carlo Method

12. Department of Mechanical Engineering Michigan State University
Summary:
• Fortran code for Monte Carlo simulations
Case Study 1: Phase diagram of a Lennard-Jones fluid
1. Fortran 77 or 90 compiler and test it
2. Download the Fortran code
3. Try to compile and run (follow the instructions)
Reading: Textbook
Chapter 3 and 5
Lecture Notes
Discussion Forum on
Angel, Examples
• Next: Molecular Dynamics method
• Compute velocity autocorrelation function (for different T, ρ, friction)
• Individual projects (2-3 slides: what is the problem, what/how to measure?)
Thursday February 3 (about 5min/presentation)
Homework 1: (P,ρ) diagram at T=2.0 and 0.9